Algebraic modular forms have been around since the work of Gross [30]. It was generally observed by him that certain reductive groups have automorphic forms that can be described in a purely algebraic way. We will motivate this for certain types of group as well as outlining the underlying theory.
After this we will see the famous correspondence of Eichler between specific spaces of elliptic cusp forms and spaces of algebraic modular forms for multi- plicative groups of definite quaternion algebras. This link is the main source of inspiration for the conjectural correspondence of Ibukiyama, which allows us to study certain spaces of Siegel modular forms algebraically too.
Most of the results in this section are well known although the proofs are often neglected. I have tried, where possible to fill in my own proofs.
3.2. Algebraic modular forms
We start with a connected reductive group G/Q with the added condition that the Lie group G(R) is connected and compact modulo center.
Recall that a choice of level structure is given by a choice of open compact subgroup K = K∞Kf ⊂ G(A) with K∞ = Z(R)K0 ⊆ G(R) and K0 maximal
compact.
Let V be (the space of) a finite dimensional algebraic representation of G, defined over a number field F . Fixing a basis of V such a representation returns for each g ∈ G a matrix ρ(g) ∈ GLn(F ) defined by polynomial equations in the
entries of g. Since V is viewed as a “weight” for our forms we need this technical assumption to avoid having ”fractional” weights.
Definition 3.2.1. The F -vector space of algebraic modular forms of level K, weight V for G is:
A(G, K, V ) = {h : G(A) → V | h(γgk) = k∞−1h(g) , ∀(γ, g, k) ∈ G(Q)×G(A)×K}.
One sees that the above definition mimics the automorphic form theory, yet it is not the best description to use computationally.
If we undo the action of the infinite component we find that Lemma 3.2.2. There is a natural isomorphism of vector spaces:
A(G, K, V ) ∼= {f : G(A) → V | f (γgk) = γf (g), ∀(γ, g, k) ∈ G(Q) × G(A) × K}. Proof. Given h ∈ A(G, K, V ) we define f : G(A) → V via f (g) = g∞h(g). The
resulting function is easily checked to satisfy the conditions required. Thus the map h 7→ f is well defined.
It is also trivial to check that this map is an invertible linear map. Hence the two spaces are naturally isomorphic.
The above lemma lets us trade invariance under G(Q) on the left into in- variance under K∞ on the right. This allows us to minimise the involvement of
the infinite place in calculations. From now on we will use this description of the space of algebraic modular forms.
Consider the adelic modular curve G(Q)\G(A)/K. Recall that it has a decomposition
G(Q)\G(A)/K ∼=a
m
Γm\HG,
where HG= G(R)/K∞ is the symmetric space attached to G. By the assump-
tion on G the symmetric space is finite. Thus the automorphic forms for such a G can be described in purely algebraic terms, since the “modular curve” is finite.
Example 3.2.3. It is easiest to see the above when G is such that G(R) is compact (e.g. special orthogonal and special unitary groups). In this case we are forced to take K∞= G(R), so that the symmetric space is a single point. It
follows that G(Q)\G(A)/K is in bijection with G(Q)\G(Af)/Kf (a set known
to be finite). For such a group it suffices to define the space of algebraic modular forms as:
A(G, Kf, V ) = {f : G(Af) → V | f (γgkf) = γf (g), ∀(γ, g, kf) ∈ G(Q)×G(Af)×Kf}.
The groups used in this thesis do not have the property that G(R) is compact. However we may still use the same argument in more generality.
Let S be the maximal split torus in the center of G. Then we have a de- composition G(R) = S(R) × G(R)1, found in the proof of Proposition 1.4 of [30]
(where G(R)1 is a certain “norm one” subgroup constructed before the proof).
Example 3.2.4. Let G be such that G(R) is compact modulo center and that the center Z(R) is a split torus. Then S(R) = Z(R) and G(R)1 ∼= G(R)/Z(R)
is maximal compact. Hence the symmetric space G(R)/Z(R)G(R)1 is a single
point. Thus in this case we can still view algebraic modular forms as functions on G(Af). We will take this stance from now on.
A natural question to ask is whether these spaces of forms are finite dimen- sional, as is the case for elliptic and Siegel modular forms. Fortunately they are and this is much easier to prove than expected. In fact, to do this we will see an explicit description of these spaces, which lends itself to computation.
Suppose we have representatives z1, z2, ..., zh∈ G(Af) for G(Q)\G(Af)/Kf.
Then it is easy to see that any f ∈ A(G, Kf, V ) is determined completely by
its values on the representatives zm. Indeed each g ∈ G(Af) generates a double
coset equal to one generated by zm for some (unique) m. Then g = γzmk for
some γ ∈ G(Q) and k ∈ Kf, so that f (g) = γf (zm).
It is now clear that the map:
φ : A(G, Kf, V ) −→ Vh
f 7−→ (f (z1), ..., f (zh))
is an embedding. However φ is not an isomorphism since the values f (zm)
cannot be chosen arbitrarily to lie in V . There is a simple restriction that can be placed on these values that does provide an isomorphism, hence an explicit description of the spaces.
Theorem 3.2.5. The map φ induces an isomorphism: A(G, Kf, V ) ∼=
h
M
m=1
3.2. Algebraic modular forms
where Γm= G(Q) ∩ zmKfzm−1 for each m.
Proof. First we see that this map is well defined. Indeed take f ∈ A(G, Kf, V ).
Then for a fixed 1 ≤ m ≤ h we can check that f (zm) ∈ V is fixed by the action
of Γmas follows.
Let γ ∈ Γmso that γ = zmkz−1m for some k ∈ Kf. Then since γ ∈ G(Q)
γf (zm) = f (γzm) = f (zmkzm−1zm) = f (zmk) = f (zm).
The map φ is clearly linear and injective and so it remains to prove surjectivity. Take (v1, v2, ..., vh) ∈ ⊕hm=1VΓm. Then we may define
f : G(Af) → V
g 7−→ γvm
whenever g ∈ G(Q)zmKf with g = γzmk for γ ∈ G(Q) and k ∈ Kf.
We first show that f is well defined, i.e. independent of the choice of γ, zm
and k. It is clear that the choice of zm is unique since z1, ..., zh form a set of
representatives for the double coset G(Q)\G(Af)/Kf.
Suppose γ, γ0 ∈ G(Q) and k, k0 ∈ K
f are such that γzmk = γ0zmk0. Then
γ−1γ0 ∈ G(Q) but also γ−1γ0 = z
mkk0−1zm−1 ∈ zmKfzm−1. Hence γ−1γ0 ∈ Γm
and since vm∈ VΓm it is now clear that γ−1γ0vm= vm. Thus γvm= γ0vmand
so f is well defined.
It remains to prove that f ∈ A(G, Kf, V ). Let γ ∈ G(Q), g ∈ G(Af) and
k ∈ Kf.
We note that if g = γ0zmk0 for some γ0 ∈ G(Q) and k0 ∈ Kf then γg =
(γγ0)zmk0 and gk = γ0zm(k0k) and so
f (γg) = (γγ0)vm= γ(γ0vm) = γf (g)
f (gk) = γ0vm= f (g)
as required.
Corollary 3.2.6. The spaces A(G, Kf, V ) are finite dimensional with
dim(A(G, Kf, V )) = h
X
m=1
dim(VΓm) ≤ h dim(V ).
Proof. This follows from the theorem since V is finite dimensional, so each VΓm
Fortunately we know more about the groups Γmwhen G is sufficiently nice.
Proposition 3.2.7. If G(R) is compact then each Γmis a finite group.
Proof. By the assumption on G(R) each Γm is discrete and compact, hence
finite (since a disjoint open cover is given by the open sets g for g ∈ G). For other groups the Γmgroups may be infinite.
Example 3.2.8. For a number field F with ring of integers OF let G =
ResF /Q(Gm). Note that at the infinite place we definitely have compactness
modulo center (since this is an abelian group).
Then for Kf = Qv-∞Ov× we find that G(Q)\G(Af)/Kf is the ideal class
group of F .
The groups Γmare all equal (since again we are in an abelian group). They
are all copies of the unit group O×F. It is known that these groups can be infinite (for example if F is a real quadratic number field).
To fix this issue Gross was able to give several equivalent conditions on a connected reductive group G that guarantee the Γm groups to be finite [30].
One such condition is that the group G(R)1 mentioned before Example 3.2.4 is
maximal compact. A simpler condition is the following:
Proposition 3.2.9. The groups Γmare finite if and only if G(Z) is finite.
Note that for the example, G(Z) = O×F and in this case the condition cap-
tures what we want.
Later we will be interested in the computation of the groups Γmbut for now
it remains to construct Hecke operators for algebraic modular forms.
We do this by first choosing an adelic point u ∈ G(Af) and decomposing the
double coset KfuKf =` r
i=1uiKf. As usual a finite number of representatives
occur. We then define a Hecke operator Tuon A(G, Kf, V ) via
Tu(f )(g) := r
X
i=1
f (gui), ∀g ∈ G(Af).
It is easy to see that this is independent of the choice of representatives uisince
they are determined up to right multiplication by Kf.
Proposition 3.2.10. Tu(f ) ∈ A(G, Kf, V ).
Proof. It is clear that if γ ∈ G(Q) then for each g ∈ G(Af):
Tu(f )(γg) = r X i=1 f ((γg)ui) = r X i=1 γf (gui) = γTu(f )(g),
3.2. Algebraic modular forms
Next note that left multiplication by Kf is a faithful action on the left cosets
uiKf. Indeed given k ∈ Kf it is clear that k(uiKf) = (kui)Kf is a left coset of
Kf in KfuKf. The group action axioms are trivial to check and faithfulness is
clear since (kum)Kf = (kun)Kf implies umKf = unKf. Thus for each k there
exists a permutation σ ∈ Sr such that (kui)Kf = uσ(i)Kf for each i.
Thus for each i we may choose kσ(i)∈ Kf such that kui= uσ(i)kσ(i). Then:
Tu(f )(gk) = r X i=1 f (gkui) = r X i=1
f (guσ(i)kσ(i)) = r
X
i=1
f (guσ(i)) = Tu(f )(g).
So Tu(f ) ∈ A(G, Kf, V ) as required.
Definition 3.2.11. The uiare often referred to as Hecke representatives and the
number r is known as the degree of the Hecke operator Tu, denoted deg(Tu).
In practice the choice of u will be of arithmetical significance (instances of this will be clear later).
We wish to find the Hecke representatives explicitly and efficiently. For this purpose a useful observation can be made when the class number is one. Proposition 3.2.12. If h = 1 then we may choose Hecke representatives that lie in G(Q).
Proof. Take any set of Hecke representatives {u1, u2, ..., ur} for Tu. If h = 1
then G(Af) = G(Q)Kf (taking the identity as representative for the double
quotient).
Thus in particular, for each i there exists γi ∈ G(Q) and ki ∈ Kf such
that ui= γiki. But then uiKf = γiKf and so we have found a set of rational
representatives: KfuKf = a uiKf = a γiKf.
From a computational perspective it is an advantage to know in advance that the Hecke representatives can be taken to be rational.
Finally we note that for G satisfying Proposition 3.2.9 there is a natural inner product on the space A(G, K, V ). This is given in Gross’ paper [30] but we shall give the rough details here.
Lemma 3.2.13. Let G satisfy the property of Proposition 3.2.9 and V be a finite dimensional algebraic representation of G, defined over Q. Then there exists a character µ : G → Gm and a positive definite symmetric bilinear form
h, i : V × V → Q such that:
for all γ ∈ G(Q).
Proof. We sketch a proof of this result. A complete argument is found in Gross’ paper.
For such a G Gross constructs a certain compact subgroup G(R)1of “norm
one” elements (see Gross p.63). This subgroup will be clear in all of our appli- cations.
By the usual averaging argument we can form a real valued G(R)1-invariant
inner product on V ⊗ R. Using this we can make a G(Q)1-invariant rational
valued form on V .
Fixing a maximal torus T lying in the center of G it is known that there exists a character χ : T 7→ Gm and a projection π : G → T such that for
γ ∈ G(R):
hγu, γvi = χ(π(γ))2hu, vi
for all u, v ∈ V ⊗ R. This is the central character of V .
One then finds the unique character µ : G → Gm such that µ|T = χ2. Then
µ satisfies hγu, γvi = µ(γ)hu, vi for all u, v ∈ V and γ ∈ G(Q).
The character µ actually takes positive values on G(R). On the adeles we get a character µ : G(A) → A×. But we may compose this with the natural projection map A×→ Q×
given by the decomposition A×= Q×R+Zˆ× to give a homomorphism µA: G(A) → Q×, giving positive values.
Proposition 3.2.14. Let G satisfy Proposition 3.2.9. Choose representatives z1, z2, ..., zh∈ G(A) for G(Q)\G(A)/Kf and fix positive definite symmetric bi-
linear form on V as in Lemma 3.2.13.
The space A(G, K, V ) has a natural inner product given by:
hf, gi = h X m=1 1 |Γm|µA(zm) hf (zm), g(zm)i, where Γm= G(Q) ∩ zmKzm−1.
Proof. The inner product axioms are clear. We show that the definition is independent of the choice of representatives zm.
Suppose y1, ..., yhis another set of representatives, ordered so that G(Q)ymK =
G(Q)zmKf. Then for each m we have γm ∈ G(Q) and km ∈ K such that